We have speculated that in Aristotle's Topics, the more elaborate theory of definition in Books VI (and in particular the architecture of the species-genus hierarchy in Book IV) grew initially from the theory of property (idion) . Some textual evidence can be adduced in favour of this (cf. 154b). There are passages in which Aristotle relaxes all the complex structural requirements on the definiens term, even simple terms are allowed to be definitions; on the other hand the remarkable passage in 132a indicates that property must have a genus + specifications structure just like definition ! How are we to understand that "terrestrial bipedial animal" indicates the essence of man but that (132a) "tame (civilized, hêmeron ) animal" or "animal gifted with knowledge" does not ? It might be tempting to view book V of the Topics, dealing with property, to be both among the oldest portions of the said work and perhaps even as having constituted an independent treatise. It is true, that definition, even difference as well as predication 'according to essence' is mentioned (and contrasted with property) in book V, but not in detailed way, rather the passages in question could easily be seen as later additions or revisions. We propose that book V be studied on its own in a more-or-less self-contained way.
To the modern reader the whole theory is highly perplexing. Not even the basic syntactical, grammatical and logical terminology is ever made fully explicit or defined clearly. As as for $A$ being predicated of $B$ 'according to essence', it baffles us. For what justification or criterion is used ?
It is very difficult and important to sort out what Aristotle means by 'subject', 'term', 'name' (onoma), 'phrase', 'predicate', 'verb', etc. These are obviously all quite different from our modern use. Most notable is the passage 132ba8-10 which distinguishes between 'name' and the 'logos' of the name. For the moment we note that by 'term' we should understand any noun phrase. In particular those noun-phrases which can be seen as having an 'extension', objects of which the noun-phrase can be predicated.
In the most basic sense, property is related to co-extensionality. A term $A$ is a property of term $B$ if both terms have the same extension, which we write $E(A) = E(B)$. But this is not all. Originally property, idion, seems to have been subordinated to pragmatic, epistemic purposes. Property was a kind of 'distinguishing mark' (perhaps not unlike linga in the Indian tradition) which served to make an object known. Thus it is natural to suppose that the property is somehow conceptually simpler than that which it is a property of. For example certain marks distinguishing certain species of plants.
So even at this stage it is clear that for Aristotle co-extensionality is a necessary but not sufficient condition for being a property (cf. also 153a in Book VII: equality is not enough for there to be definition). Thus being a property is also intensional. If we take the sentence "$X$ is a property of $A$" then we cannot substitute co-extensional terms for $X$ salva veritate. Indeed Aristotle's predicate of terms ' being better known' (see 129b-130a) is intensional. We denote this situation by $A\prec B$ to mean that the term $B$ is better known than $A$. $\prec$ is a transitive. We write $A\sim B$ if $A$ and $B$ are simultaneously known (this can be defined as $A\nprec B\& B\nprec A$). This can only involve the respective concepts and not the extensions. If we denote by $A^\circ$ the contrary of the term $A$ then we have the following rule in 131a: $ A \sim A^\circ$.
One of the most interesting aspect of the theory of property (contained in the first four sections of Book V) is the series of syntactic-semantic (intensional) conditions imposed on the property term. The theory involves implicit syntactic-grammatical notions which unfortunately are either lost or never made explicit in surviving texts. Nevertheless the theory seems be surprisingly modern. Here are, in a modernized language, some of the conditions. For $B$ to be a property of $A$ it is required that:
i) the term $A$ does not occur in $B$ (non-circularity) (131a)
ii) the term $B$ is different from the term $A$ (non-identity) (135a)
iii) the term $B$ is not a conjunction $B_1 \& B_2$ in which $B_1$ or $B_2$ are already properties of $A$ (irreducibility)
iv) $B$ is not a universal predicate or does not make use of universal predicates (non-redundancy)(130b)
v) the term $B$ does not make use of terms lesser known than $A$ (epistemic priority)(129b)
vi) the term $B$ does not have repeated subterms (including via anaphora) (130b)
Perhaps iii) can be replaced with
iii)' the term $B$ does not strictly contain a term $B'$ which is a property of $A$.
As a corollary of v) we obtain that: $B$ cannot contain $A^\circ$ (131a).
Note of these conditions would we rejected, for instance, by a modern mathematician who wishes to "characterize" (but not necessarily define) a class of objects. Even if the 'better known' relation is questionable, characterizations involving theory disproportionately more elaborate and distant than those of the the original class of objects will not always be welcomed (though this discussion needs to be refined).
We now investigate the topics involved with the relation between $B$ being the property of $A$ and the relationship between $B$ and terms $C$ in types of fundamental relation to $A$. For instance in 131a condition i) is actually strengthened to i)' the term $B$ does not include terms $C$ which are contained in $A$ (seemingly in the sense of species). We denote this inclusion relation by $C\subset A$.
Thus we have the topic: $\exists C. C\subset A \& C\notin B \,\rightarrow\, \neg I(A,C)$. Here $\in$ and $\notin$ represent simple predication and $I(A,B)$ the circumstance of $B$ being a property of $A$ (132a).
But what about individuals which are part of the extension of $A$, $E(A)$, how do they enter in topics of property ? At first sight individuals would seem to have little relationship with properties of the term to whose extension they belong. For indeed one man is not distinguished from another man on account of such a property (but cf. Nietzsche's wenn du eine Tugend hast, und es deine Tugend ist, so hast du sie mit niemandem gemeinsam). Of course just as in the topic above, the property of the species must be predicates of each individual in the extension of $A$. But does it make sense to speak of property here ? How does one extrapolate from predication of individuals to a property of their species ? Aristotle introduces the construction: '$B$ is said of all individuals $a \in E(A)$ qua $A$' (132b). How are we to interpret this ? Why would not a single individual suffice ? What about the distributivity of the 'all' ?
But first a remark on 132b, on the 'onoma' and the 'logos'. Perhaps these refer simply to the subject and to what is proposed as the property of a subject. Thus the destructive topic would read: $\exists X. ((X\in A)\& (X\notin B) \vee (X \notin A) \& (X\in B))\rightarrow \neg I(A,B)$. The constructive topics seems however insufficient, for this would hold for definitions.
The following passage (133b) is interesting yet quite difficult to interpret: (W.A. Pickard translation):
Next, for destructive purposes, see if the property of things that are the same in kind as the subject fails to be always the same in kind as the alleged property: for then neither will what is stated to be the property of the subject in question. Thus (e.g.) inasmuch as a man and a horse are the same in kind, and it is not always a property of a horse to stand by its own initiative, it could not be a property of a man to move by his own initiative; for to stand and to move by his own initiative are the same in kind, because they belong to each of them in so far as each is an 'animal' .
The subtext here seems to be that if $C_1$ and $C_2$ are species of a genus $G$ and $C_1$ has attribute $A_1$ then there is a canonically morphism which applied to $A_1$ yields an attribute $f_C(A_1)$ of $C_2$. There seems to be analogy involved here. But why is 'moving' to man analogous to 'standing' for horse ?
Book V is a good place to focus on for the study of opposites (treated is section/part 6). Aristotle's theory of opposite is difficult to fathom from a modern perspective. Opposition is not classical or intuitionistic negation (Aristotle has in fact a theory of the positive and negative). Rather it is a algebraic-like operator defined on the class of terms; and in fact there are many distinct species of opposition. This suggests that genera have in general automorphisms acting on species. For the present we assume that our notation $T^\circ$ refers to contraries. We have $T^{\circ\circ} = T$. We interpret terms 'not having an opposite' as having $T^\circ = T$. A striking feature is that $E(T^\circ)$ is not the complement of $E(T)$. This is substantiated by 135b in which the contrary of 'the highest good' is 'the worst evil'. The following is the fundamental topic for property and contraries:
\[ I(A,B) \leftrightarrow I(A^\circ, B^\circ) \]
We denote Aristotle's 'modern' notion of term-negation by $n(T)$. The associated topics are not surprising (136a). For instance $I(A, n(B)) \rightarrow \neg I(A,B)$ and $I(A,B) \rightarrow \neg I(A,n(B))$.
Part 8 of book V deals with degrees of terms as well as with what might be called probabilistic logic. The aspect of degree of a term seems very much to act like an operator on terms. Thus we can denote 'more','most', 'less' and 'least' $T$ by $T^+, T^\triangle, T^-, T^\bigtriangledown$. If a term does not admit degree then we assume that $T^+ = T$, etc. The related topics can be condensed into
\[ I(A,B) \leftrightarrow I(A^+, B^+) \leftrightarrow I(A^-, B^-) \leftrightarrow I(A^\triangle, B^\triangle) \leftrightarrow I(A^\bigtriangledown, B^\bigtriangledown)\]
It does not seem that degree has directly comparative aspect. Rather the degree is taken as relative to some implicit absolute standard. But for probability degree the situation is otherwise. Let us write $\mathbb{P}I(A,B) > \mathbb{P}I(A'.B')$ for 'it is more likely for $B$ to be the property of $A$ then $B'$ to be the property of $A'$. Then Aristotle's rather strange supposition is that
\[ \mathbb{P}I(A,B) > \mathbb{P}I(A'.B')\,\&\, \neg I(A,B) \rightarrow \neg I(A',B') \]
In 138a we see variants on the topics corresponding to the cases in which $A = A'$ and $B= B'$. Then follows topics relating to equal likelihood which can be reduced to
\[ \mathbb{P}I(A,B) = \mathbb{P}I(A'.B') \rightarrow ( I(A,B) \leftrightarrow I(A',B')) \]
and cases in which $A= A'$, etc.
These topics are strange because Aristotle does not state clearly the intrinsic relationship between $A$ and $A'$ or $B$ and $B'$. The following difficult passage seems relevant:
The rule based on things that are in a like relation' differs from the rule based on attributes that belong in a like manner,' because the former point is secured by analogy, not from reflection on the belonging of any attribute, while the latter is judged by a comparison based on the fact that an attribute belongs.
We have deliberately refrained from discussing important modal and temporal aspects of the 'logic' of book V of the Topics. Book V ends with the curious rejection of property given by superlatives - surely not the same superlative as $T^\triangle$ but rather something like a modifier of a noun, something of the form 'the Xest Y'. Aristotle basically says that such a definite description will not denote uniquely for different states of affairs.
The topics involving analogy deserve special study; also those involving a term being identically related to other terms (137a). But what is analogy ? An isomorphism between the structure of two different structures. We could compare this with the Indian system of Nyâya: the Topics invokes strongly a 'case-based' logic according to the interpretation of Ganieri. We have $Fb$ and $Gb$ as well as $Fa$ and we want to argue that $Ga$. This is based on analogy, or relative similarity of $a$ and $b$,
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